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The standard normal distribution, a key concept in statistics and probability theory, is a special case of the normal distribution. It's a continuous probability distribution that is symmetric around its mean, which is always zero, and with a variance of 1. This particular distribution has significant importance due to its role in the central limit theorem, which states that the sum of independent and identically distributed random variables tends toward a normal distribution, regardless of the original distribution of the variables.

A vital property of the standard normal distribution is that any normal distribution can be transformed into a standard normal one through a process known as normalization. One can use the Z-score, a measure that indicates the number of standard deviations an element is from the mean. The formula for the standard normal distribution's probability density function (PDF) is crucial for performing various probabilistic calculations.

The expected value, or mean, of a random variable gives a measure of the central tendency of the distribution of that variable. It's a fundamental concept in probability and statistics that provides a baseline expectation for the outcome of a random process. For a standard normal distribution, the expected value is 0. This is because the distribution is symmetric around the mean, with equal probabilities for values above and below the mean.

Calculating the expected value involves summing up all possible values of the random variable, each weighed by its probability. These probabilities come from the probability density function. In many cases, expected values can be used to identify the 'center of mass' of a probability distribution, which is a critical point in assessing the behavior of random variables.

Variance is a measure of the spread of a probability distribution and signifies the degree of variation or spread in a set of random variables. The variance of a standard normal distribution is always 1, indicating that the typical deviation from the mean is one standard deviation. It's an invaluable concept when it comes to understanding the reliability of the mean.

Mathematically, variance is the expected value of the squared deviations from the mean, denoted as \( Var(X) \). It gives us an idea of how much the values in the random process differ from the expected value. A low variance indicates that the data points tend to be close to the mean, whereas a high variance suggests a wider range of values. In the context of standard normal distribution, since the expected value is 0 and the variance is 1, most values lie between -1 and 1.

The probability density function (PDF) is a function that describes the likelihood of a random variable taking on a given value. In the case of continuous random variables like those from a standard normal distribution, the PDF provides the probabilities over a continuous range of values.

For standard normal variables, the PDF has a very specific bell-shaped curve known as the Gauss curve. The formula for the PDF of a standard normal variable is \( f(z) = (1 / \sqrt{2\pi}) e^{-z^2/2} \), where \( z \) is the value of the random variable. This function is crucial for finding probabilities for intervals and for defining the expected value of functions of the random variables, such as in the moment-generating function.